Cahn transition critical point

Fig. 55. Schematic phase diagram of a binary mixture with an unmixing transition in the bulk (miscibility gap from composition 4 (7 ) to cJoixOT) ending in a critical point Tc, . ,) and a first-order wetting transition at Tv at the surface of the mixture and a wall. For T > rw, a (thick) layer of concentration with the other branch of the coexistence curve, cSx(T ) is adsorbed at the wall. The prevvetting line ending in a surface critical paint Tcs is also shown. After Cahn (1977).

The existence of such a transition—both when is and when it is not stable in bulk—was predicted by Cahn. He assumed the proximity of a critical point of Py (or aP) phase equilibrium, and then referred to the transition from non-wetting to wetting of the ay interface by a bulk or incipient p phase as critical-point wetting. We paraphrase here his argument that such a transition is to be expected in the neighbourhood of, say, the Py critical point. 

The important conclusion from this argument is that if a Cahn transition in the ay interface occurs near a y (or, equally well, an aP) critical point, the states in which p spreads at the ay interface are those which are nearer the critical point, while those in which does not spread are those which are further. That was brilliantly verified in the experiments of Moldover and Cahn. As a corollary of great practical importance, we note that sufficiently near to a or a critical point, the... 

In Fig. 8.11, which we have adapted from Cahn and from Teletzke er al., we show the temperature (T) vs composition (x) coexistence curve for the equilibrium of the phases B and y (two liquids, say), while these are also in equilibrium widi a third phase, a, which is not shown in the diagram (a vapour phase, say, or a solid boundary). The By critical point is at C. The points marked y and B tmd shown connected by a tieline are a general pair of equilibrium y and B phases. The tieline labelled P marks the Cahn transition in the three-phase (apy) region, and corresponds to P in Hg. 8.10. In the three-phase region above P, that is, dcmr to the critical point C, the ay interface is wetted by B, below P it is not. 

Tbe point C is the critical point of this interfiuaal phase transition, also predicted by Cahn. Along P C the two alternative structures of the ay interface are of equal tension. As C e approached, those two equally stable but dtetinct structures become gradually more alike, in die way we saw in our description of the critical points of general first-order interfa-dal phase transitions. At C they have become identical. 

While the situation of a liquid in coexistence with its vapor phase is related to systems familiar from everyday life, the same considerations apply to a coexisting binary liquid mixture in contact with a third phase (vapor or solid), as first shown in a experiment by Moldover and Cahn2. For a binary liquid mixture replace the indices 1, v in eq. 1 with a and p, the coexisting phases of the liquid-A/liquid-B mixture. Cahn showed in an elegant argument, that a wetting transition always has to exist as a bulk critical point is approached along the coexistence curve. 

Fig. 14.a Composition-depth cf)(z) profiles near the surface (at z=0) of a binary mixture at bulk concentration bi at lower (T—>TW ) and upper (T—>TW+) limit of the first order wetting transition point b,c Cahn constructions with trajectories -2kV< ) plotted for profiles cf)(z) with decreasing (solid lines) and increasing (dashed lines) slopes. Surface boundary condition (Eq. 26) is met at points (marked by ) where surface energy derivative (-dfs/d< ))s (idotted line) intersects trajectories -2kV< > at concentrations reached at the surface. Cahn plot b corresponding to the first order transition depicted in a Cahn construction c typical for a critical wetting trajectories -2kV< > with larger extrema correspond to temperatures T[Pg.37]

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